Two-level Parallel Computation for Approximate Inverse with AISM Method

Transcription

1 Vol 49 No (June 2008) AISM AISM SGI Orgn 2400 MPI PE16 PE Two-level Parallel Computaton for Approxmate Inverse wth AISM Method Lnje Zhang, 1 Kentaro Morya 2 and Takash Nodera 3 In ths paper, we propose a novel strategy for parallel precondtonng of large scale lnear systems by means of a two-level approxmate nverse technque wth AISM method Accordng to the numercal results on an orgn 2400 by usng MPI, the proposed parallel technque of computng the approxmate nverse makes the speedup of about tmes wth 16 processors Graduate School of Scence and Technology, Keo Unversty 2 Faculty of Scence and Technology, Aoyama Gakun Unversty 3 Faculty of Scence and Technology, Keo Unversty Ax = b, A R n n, x, b R n (1) (1) (1) Saad 8) [pp ] (1) M R n n (1) A AMy = b, x = My (2) M A eg Chow 3) [pp ] Bru 2) AISM 1 AISM A A 1 α 1 I n α 2 UΩ 1 V T, (α >0) (3) I n R n n U V u k v k (4) k 1 u k = x k =1 =1 (v ) k αr u, k 1 y T k v k = y k u v αr (3) Ω r k =1+(v k ) k /α Bru 2) (3) α (4) x k y k α =15 A, x k = e k, y k =(a k αe k ) T e k R n I n k a k A k U V U V AISM 2) u k v k 6) Nak 7) AISM (4) 2164 c 2008 Informaton Processng Socety of Japan

2 2165 AISM 2 5) 8) [pp75 84] 2 CPU 1),4) 2 AISM MPI SGI Orgn ) A 1 B 1 P T AP = (5) A p B p C 1 C p A s P (5) A A 4) Benz 1) AINV 2 2 AISM P T AP A 1 1 E 1 I 1 (P T AP ) 1 = A 1 p E p I p S 1 F 1 F p I s E = A 1 B S 1 F = C A 1 S = A s p C =1 A 1 B A s I I s A A s A 1 S 1 M M s P T AP (P T AP ) 1 M 1 Ē 1 M p Ē p M s I 1 I p F 1 Fp I s M A 1 M s S 1 Ē = M B M s F = C M S = A s p C =1 M B C M B S 2 (6) 2 P T AP 1 M A 1 =1,,p 2 M s S 1 Ē F Ē F Ē x = M (B (M s x)) F x = C (M x) A =1,,p S P T AP p Fg 1 1 for =1,, s 2 compute M A 1 3 compute S = C M B 4 end for 5 compute S = A s p S j=1 j 1 6 compute M s S 1 2 Algorthm of the two level computaton (6) Vol 49 No (June 2008) c 2008 Informaton Processng Socety of Japan

3 2166 AISM 2 PE 1 PE 2 PE p 1 compute M 1 A 1 1 compute M 2 A 1 2 compute M p A 1 p 2 compute S 1 = C 1 M 1 B 1 compute S 2 = C 2 M 2 B 2 compute S p = C p M p B p 3 receve S, ( =2,,p) from other PEs send S 2 to PE 1 send S p to PE 1 4 compute S = A s p S j wat wat j=1 5 compute M s S 1 wat wat 6 send M s to other PEs receve M s from PE1 receve M s from PE1 Fg Steps of two level parallel computaton p A A s p A S p PE M =1,,p S C M B =1,,p PE M =1,,p C M B =1,,p PE x R n (x 1,x 2,,x p,x s ) T x x s PE x 1,x 2,,x p,x s A 1,A 2,,A p,a s P T AP A 1 B 1 A p B p C 1 C p A s x 1 x p x s = A 1 x 1 + B 1 x s A p x p + B p x s p =1 C x + A s x s A x + B x s =1,,s C x =1,,s M M 1 Ē 1 I 1 x 1 M 1 x 1 + Ē1z s = M p Ē p I p x p M p x 1 + Ēpz s M s F 1 Fp I s x s M s z s z s = p F =1 x + x s F x =1,,s M x =1,,s α = p <x =1,y > + <x s,y s > < x,y > =1,,s 4 MPI SGI Orgn 2400 CPU 300 MHz 16 Vol 49 No (June 2008) c 2008 Informaton Processng Socety of Japan

4 2167 AISM 2 MPI-12 C 1CPU 1PE Karyps 5) pmets 1 GMRES(m) 9) 1 GMRES(m) r k 2 / b r k k GMRES(m) GMRES(m) 1 1 m 50 MPI Wtme() 2 AISM α Bru 2) α =15 A U V 01 GMRES(m) Ω=[0, 1] [0, 1] {( u xx u yy + D y 1 ) ( u x + x 1 )( x 2 ) } u y 43π 2 u = G(x, y), u(x, y) Ω =1+xy G(x, y) u =1+xy Ω ,864 Dh 2 7 h h =1/193 1 PE 2 AISM 3,79965 p (5) 1 pmets A =1,,pp A A s p =2 A 1 A 2 A s 18,239 18, p =12 A s A 1 2 Table 1 Computaton tme and Speedup of 2 level (parallel) computaton for an approxmate nverse 2 2 p 2 1, p 14 A s A p =16 A 2,150 A s 2,460 1 PE 2 PE PE PE p p =14 p 14 A s A S 2 GMRES(50) (7) 2 Tpre I Tgmres GMRES(50) Tgmres Tpre T total T total = Tpre + Tgmres E total 1 PE 2 2 p =14 p = GMRES(m) A s GMRES(m) Vol 49 No (June 2008) c 2008 Informaton Processng Socety of Japan

5 2168 AISM Table 2 GMRES(50) Numercal results of GMRES(50) p Tpre I Tgmres T total E total ,195 5, , ,297 1, , ,006 1, , ,207 1, , ,595 1, , , , ,395 1, , AISM SGI Orgn AISM 1 1) Benz, M, Marín,JandTůma, M: A Two-level Parallel Precondtoner Based on Sparse Approxmate Inverse, IMACS Seres n Computatonal and Appled Mathematcs, Vol5, pp , IMACS (1999) 2) Bru, R, Cerdán, J, Marín, J and Mas, J: Precondtonng Sparse Nonsymmetrc Lnear Systems wth the Sherman-Morrson Formula, SIAM J Sc Comput, Vol25, No2, pp (2003) 3) Chow, E and Saad, Y: Approxmate Inverse Precondtoners va Sparse-sparse Iteratons, SIAM J Sc Comput, Vol19, No3, pp (1998) 4) Heath, MT, Ng, E and Peyton, BW: Parallel Algorthms for Sparse Lnear Systems, SIAM Revew, Vol33, No3, pp (1990) 5) Karyps, G and Kumar, V: A Fast and Hgh Qualty Multlevel Scheme for Parttonng Irregular Graphs, SIAM J Sc Comput, Vol20, No1, pp (1998) 6) Sherman-Morrson Vol48, No3, pp (2007) 7) Nak, VK: A Scalable Implementaton of the NAS Benchmark BT on Dstrbuted Memory Systems, IBM Systems Journal, Vol34, No2, pp (1995) 8) Saad, Y: Iteratve Methods for Sparse Lnear Systems, PSW Publshng Co, Boston (1966) 9) Saad, Y and Schultz, MH: GMRES: A Generalzed Mnmal Resdual Algorthm for Solvng Nonsymmetrc Lnear Systems, SIAM J Sc Statst Comput, Vol7, No3, pp (1986) ( ) ( ) 1997 & IT L A TEX SIAM Vol 49 No (June 2008) c 2008 Informaton Processng Socety of Japan